Prime Minister Menachem Begin today formally requested the head of the Israeli Supreme Court to establish a state judicial board of inquiry into the Beirut massacre, putting into motion yesterday's Cabinet decision to authorize the independent investigation.

Begin's action came as tensions here seemed to ease after almost two weeks of turmoil. It was a day in which the Cabinet did not meet, the Knesset (parliament) was not due back in session until mid-October, and the Foreign Ministry for the first time in weeks said it was pleased by a statement coming out of Washington.

Yesterday's decision to investigate the extent of Israel's role in the massacre of Palestinian civilians by Lebanese Christian militiamen was widely acclaimed in the Israeli press. But in a noisy echo of the rancor and consternation here since the massacre, the government's reluctance to allow the inquiry until yesterday, nearly two weeks after the killings, continued to come under severe criticism.

Begin asked formally for the establishment of the inquiry board in a letter to the Supreme Court chief justice, Yitzhak Kahan. Kahan is expected to appoint himself or his predecessor, retired chief justice Moshe Landau, to head the panel.

The three-member board is expected to hear testimony from Begin, Defense Minister Ariel Sharon and other top political and military leaders in an investigation that is predicted to last a month to six weeks.

The panel is merely a fact-finding body with no judicial powers. But its findings on the decision to send Lebanese Christian militia units into the Shatila and Sabra refugee camps and the failure to halt the massacre until about 36 hours had elapsed could have a strong impact on the future of the Begin government and particularly the personal fortunes of Sharon, who remains the object of the most intense criticism.

Under Israeli law, public comments and news stories dealing directly with the facts surrounding the massacre will be restricted during the investigation, a respite that government and military leaders will undoubtedly welcome after the flood of damaging stories in the press here in the last week about the Israeli role in the events.

The easing of tension appeared also to apply to the recently troubled relationship with Washington. The Foreign Ministry said it was pleased by President Reagan's comments at a news conference last night, calling the president's reiteration of support for Israel a "positive step."

A spokesman said Reagan's remarks "stated very clearly what we believe still to be the case -- that is the positive character of the basic relationship between the two countries."

Begin received another bit of good news today in the form of a public opinion poll conducted by the Dahaf Institute for Monitin magazine. The poll, taken last week at the height of the storm over the massacre and the government's refusal to establish a state board of inquiry, showed that Begin's popularity has been eroded by the incident but that he and his governing Likud Bloc remain in a commanding position.

The number of Israelis who say they are satisfied with Begin's performance as prime minister declined from 82 percent before the massacre to 72 percent, while those who say they are dissatisfied increased from 15 percent to 27 percent, the poll results showed.

According to Mina Zemach, who directed the poll, the massacre cost Begin the surge in support he had enjoyed because of the war in Lebanon, but it has not altered fundamentally the prime minister's dominant position in Israeli politics.

Sharon's popularity suffered even more, but he still commands the support of 64 percent of the public, according to the poll.

Sharon's future appears to be the most cloudy as a result of the massacre. The independent newspaper Haaretz today renewed its call for the defense minister's ouster, while the English-language Jerusalem Post said in an editorial that in any other country both Begin and Sharon would have resigned immediately after the massacre in Israeli-occupied West Beirut became known.

The strongest threat to Sharon appears to come from within the Army. Today, a petition signed by about 1,000 Army reserve officers and soldiers asking not to be ordered to serve in Lebanon was presented to the Defense Ministry. The petition drive was organized by a group of antiwar activists in the military and seemed to indicate the growing disillusionment with Sharon's leadership within the military.

Dramatic steps such as resignations and firings, however, are not expected here until after the inquiry board reports its findings, if at all. Ronnie Medzini, a former opposition Labor Party spokesman and longtime observer of Israeli politics, said in an interview that he expects the turmoil in the country to continue but "on a somewhat lower key."

The Labor Party is already attempting to woo away from Begin the members of the National Religious Party, most of whose leaders are "pretty shook up" over the massacre, Medzini said.

Begin today appeared before a closed meeting of the Knesset Foreign Affairs and Defense Committee, where he reportedly said that as prime minister he would take full responsibility for the events in Beirut and not blame others for any Israeli failures.

According to Israel Radio, Begin was in his normally defiant mood before the committee, accusing his political opponents of attempting to undermine the goverment. Many here believe that if pressure is maintained on Begin, he eventually will be forced to dismiss Sharon. A Begin associate noted, however, that Sharon's firing could bring the resignation of five or six other members of the government on the far right.

In effect, he said, firing the defense minister could bring the downfall of the Begin government.

Meanwhile, the military command today issued a statement in response to published reports in The Washington Post and elsewhere of looting by Israeli soldiers in Lebanon. Without giving specifics, the military command said there have been "a few cases" of looting by Israeli soldiers during the war, resulting in courts-martial and severe punishment for those found guilty.